The England Under-21 star has made an exceptional start to the Championship season and has already claimed nine goals.
A host of Premier League clubs have been linked with the youngster, most recently Johnson's hometown club Sunderland.
But Boro chief executive Keith Lamb insists that their North East rivals have not been in touch.
"We have had no approach from Sunderland, nor would we welcome one," insisted Lamb.
Lamb maintains Boro want Johnson to remain, although thus far the 22-year-old has shunned efforts to prolong his stay at The Riverside Stadium.
"Adam is a Middlesbrough player and we want him to stay," he said.
Johnson's current deal is due to expire in the summer, and he will be free to join any club he wants come the end of the season - although Boro would be due compensation as he is under 24.